Chapter 12.3, Problem 99AYU

To determine

To calculate: The payable amount for the stock, if a stock pays annual dividend of $\$4.00$ and the dividend has been increased $3\%$ per year and an annual rate of return is $9\%$, by using the present value of an infinite stream of payments:

$\text{Price}=P+P\cdot \frac{1+i}{1+r}+P.{\left(\frac{1+i}{1+r}\right)}^2+P.{\left(\frac{1+i}{1+r}\right)}^3+\cdot \cdot \cdot$

Answer to Problem 99AYU

Solution:

One needs to pay approximately $\$72.67$ for the stock.

Explanation of Solution

Given information:

A stock pays annual dividend of $\$4.00$, the dividend has been increased $3\%$ per year, and an annual rate of return is $9\%$.

Formula used:

If $\left|r\right|<1$, the infinite geometric series ${\displaystyle \sum _{k=1}^{\infty }{a}_1{r}^{k-1}}$ converges. Its sum is ${\displaystyle \sum _{k=1}^{\infty }{a}_1{r}^{k-1}}=\frac{a_1}{1-r}$

Calculation:

The present value of an infinite stream of payments is $\text{Price}=P+P\cdot \frac{1+i}{1+r}+P.{\left(\frac{1+i}{1+r}\right)}^2+P.{\left(\frac{1+i}{1+r}\right)}^3+\cdot \cdot \cdot$.

Here, $P=4.00$, $i=3\%$, $r=9\%$

By substituting the above values in the present value of an infinite stream of payments,

$\Rightarrow \text{Price}=4+4\cdot \frac{1+0.03}{1+0.09}+4.{\left(\frac{1+0.03}{1+0.09}\right)}^2+4.{\left(\frac{1+0.03}{1+0.09}\right)}^3+\cdot \cdot \cdot$

$\Rightarrow \text{Price}=4+4\cdot \frac{1.03}{1.09}+4.{\left(\frac{1.03}{1.09}\right)}^2+4.{\left(\frac{1.03}{1.09}\right)}^3+\cdot \cdot \cdot$

This is an infinite geometric series with first term $a=4$ and common ration $r=\frac{1.03}{1.09}$

By using the geometric series formula, ${\displaystyle \sum _{k=1}^{\infty }{a}_1{r}^{k-1}}=\frac{a_1}{1-r}$

$\Rightarrow \text{Price}=\frac{4}{1-\frac{1.03}{1.09}}$

By simplification,

$\Rightarrow \text{Price}\approx 72.67$

Thus, one needs to pay approximately $\$72.67$ for the stock.